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hp 49g+ graphing calculator user’s manual H Edition 2 HP part number F2228-90001.
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Preface You have in your hands a compact symbolic and numerical computer that will facilitate calculation and mathematical analysis of problems in a variety of disciplines, from elementary mathematics to advanced engineering and science subjects. This manual contains examples that illustrate the use of the basic calculator functions and operations.
Page TOC-1 Table of Contents Chapter 1 – Getting Started , 1-1 Basic Operations , 1-1 Batteries, 1-1 Turning the calculator on and off, 1-2 Adjusting the display contrast, 1-2 Contents of the calcul.
Page TOC-2 Creating algebraic expressions, 2-4 Using the Equation Writer (EQW) to create expressions , 2-5 Creating arithmetic expressions, 2-5 Creating algebraic expressions, 2-8 Organizing data in t.
Page TOC-3 Unit conversions, 3-14 Physical constants in the calculator , 3-14 Defining and using functions , 3-16 Reference , 3-18 Chapter 4 – Calculations with complex numbers , 4-1 Definitions , 4.
Page TOC-4 The PARTFRAC function, 5-11 The FCOEF function, 5-11 The FROOTS function, 5-12 Step-by-step operations with polynomials and fractions , 5-12 Reference , 5-13 Chapter 6 – Solution to equat.
Page TOC-5 Chapter 8 – Vectors , 8-1 Entering vectors , 8-1 Typing vectors in the stack, 8-1 Storing vectors into variables in the stack, 8-2 Using the Matrix Writer (MTRW) to enter vectors, 8-2 Sim.
Page TOC-6 Solution with the inverse matrix, 9-10 Solution by “division” of matrices, 9-10 References , 9-10 Chapter 10 – Graphics , 10-1 Graphs options in the calculator , 10-1 Plotting an expr.
Page TOC-7 Chapter 14 – Differential Equations , 14-1 The CALC/DIFF menu , 14-1 Solution to linear and non-linear equations , 14-1 Function LDEC, 14-2 Function DESOLVE, 14-3 The variable ODETYPE, 14.
Page TOC-8 Chapter 17 – Numbers in Different Bases , 17-1 The BASE menu , 17-1 Writing non-decimal numbers , 17-1 Reference , 17-2 Chapter 18 – Using SD cards , 18-1 Storing objects in the SD card.
Page 1-1 Chapter 1 Getting started This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aimed at familiarizing yourself with the basic operations and settings before actually performing a calculation.
Page 1-2 b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facing up. c. Replace the plate and push it to the original place. After installing the batteries, press [ON] to turn the power on. Warning: When the low battery icon is displayed, you need to replace the batteries as soon as possible.
Page 1-3 For details on the meaning of these specifications see Chapter 2 in the calculator’s user’s guide. The second line shows the characters { HOME } indicating that the HOME directory is the current file directory in the calculator’s memory.
Page 1-4 and Chapter 2 and Appendix L in the user’s guide for more information on editing) @VIEW B VIEW the contents of a variable @@ RCL @@ C ReCaLl the contents of a variable @@STO@ D STOre the contents of a variable ! PURGE E PURGE a variable CLEAR F CLEAR the display or stack These six functions form the first page of the TOOL menu.
Page 1-5 the blue ALPHA key, key (7,1) , can be combined with some of the other keys to activate the alternative functions shown in the keyboard. For example, the P key, key(4,4) , has the following s.
Page 1-6 ~p ALPHA function, to enter the upper-case letter P ~„p ALPHA-Left-Shift function, to enter the lower-case letter p ~…p ALPHA-Right-Shift function, to enter the symbol π Of the six functions associated with a key only the first four are shown in the keyboard itself.
Page 1-7 Press the !!@@OK#@ ( F ) soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. Operating Mode The calculator offers two operating modes: the Algebraic mode, and the Reverse Polish Notation ( RPN ) mode.
Page 1-8 1./3.*3. ————— /23.Q3™™+!¸2.5` After pressing ` the calculator displays the expression: √ (3.*(5.-1/(3.*3.))/23.^3+EXP(2.5)) Pressing ` again will provide the following value.
Page 1-9 different levels are referred to as the stack levels , i.e., stack level 1, stack level 2, etc. Basically, what RPN means is that, instea d of writing an operation such as 3 + 2, in the calculator by using 3+2` we write first the operands, in the proper order, and then the operator, i.
Page 1-10 5 . 2 3 23 3 3 1 5 3 e + ⋅ − ⋅ 3` Enter 3 in level 1 5` Enter 5 in level 1, 3 moves to level 2 3` Enter 3 in level 1, 5 moves to level 2, 3 to level 3 3* Place 3 and multiply, 9 appears in level 1 Y 1/(3 × 3), last value in lev.
Page 1-11 (12 significant digits).”To learn more abo ut reals, see Chapter 2 in this guide. To illustrate this and other number formats try the following exercises: • Standard format : This mode is the most used mode as it shows numbers in the most familiar notation.
Page 1-12 Press the !!@@OK#@ soft menu key to complete the selection: Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Notice how the number is rounded, not truncated. Thus, the number 123.4567890123456, for this setting, is displayed as 123.
Page 1-13 This result, 1.23E2, is the calculator’s version of powers-of-ten notation, i.e., 1.235 × 10 2 . In this, so-called, scientific notation, the number 3 in front of the Sci number format (shown earlier) represents the number of significant figures after the decimal point.
Page 1-14 • Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with such notation.
Page 1-15 • Grades : There are 400 grades ( 400 g ) in a complete circumference. The angle measure affects the trig functions like SIN, COS, TAN and associated functions. To change the angle measure mode, use the following procedure: • Press the H button.
Page 1-16 Selecting CAS settings CAS stands for C omputer A lgebraic S ystem. This is the mathematical core of the calculator where the symbolic mathematical operations and functions are programmed. The CAS offers a number of settings can be adjusted according to the type of operation of interest.
Page 1-17 options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow options above). • After having selected and unselected all the options that you want in the CAS MODES input form, press the @@@OK@@@ soft menu key.
Page 1-18 The calculator display can be customized to your preference by selecting different display modes. To see the opti onal display settings use the following: • First, press the H button to activate the CALCULATOR MODES input form.
Page 1-19 ( D ) to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0:system 8 is selected. This is the default value of the display font.
Page 1-20 Selecting properties of the Stack First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , twice, to get to the Stack line.
Page 1-21 Selecting properties of the equation writer (EQW) First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , three times, to get to the EQW (Equation Writer) line.
Page 2-1 Chapter 2 Introducing the calculator In this chapter we present a number of basic operations of the calculator including the use of the Equation Writer and the manipulation of data objects in the calculator. Study the examples in this chapter to get a good grasp of the capabilities of the calculator for future applications.
Page 2-2 Notice that, if your CAS is set to EXACT (see Appendix C in user’s guide) and you enter your expression using integer numbers for integer values, the result is a symbolic quantity, e.g., 5*„Ü1+1/7.5™/ „ÜR3-2Q3 Before producing a result, you will be asked to change to Approximate mode.
Page 2-3 To evaluate the expression we can use th e EVAL function, as follows: µ„î` If the CAS is set to Exact , you will be ask ed to approve changing the CAS setting to Approx .
Page 2-4 This expression is semi-symbolic in the sense that there are floating-point components to the result, as well as a √ 3. Next, we switch stack locations [using ™ ] and evaluate using function NUM, i.
Page 2-5 Entering this expression when the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise. For additional information on editing algebraic expressions in the calculator’s display or stack see Chapter 2 in the calculator’s user’s guide.
Page 2-6 The cursor is shown as a left-facing key. The cursor indicates the current edition location. For example, for the cursor in the location indicated above, type now: *„Ü5+1/3 The edited expression looks as follows: Suppose that you want to replace the quantity between parentheses in the denominator (i.
Page 2-7 The expression now looks as follows: Suppose that now you want to add the fraction 1/3 to this entire expression, i.e., you want to enter the expression: 3 1 ) 2 5 ( 2 5 5 2 + + ⋅ + π Firs.
Page 2-8 Creating algebraic expressions An algebraic expression is very similar to an arithmetic expression, except that English and Greek letters may be included. The process of creating an algebraic expression, therefore, follows the same idea as that of creating an arithmetic expression, except that use of the alphabetic keyboard is included.
Page 2-9 Also, you can always copy special characters by using the CHARS menu ( …± ) if you don’t want to memorize the keystroke combination that produces it. A listing of commonly used ~‚ keystroke combinations is listed in Appendix D of the user’s guide.
Page 2-10 Variables Variables are similar to files on a computer hard drive. One variable can store one object (numerical values, algebraic expressions, lists, vectors, matrices, programs, etc).
Page 2-11 To unlock the upper-case locked keyboard, press ~ Try the following exercises: ³~~math` ³~~m„a„t„h` ³~~m„~at„h` The calculator display will show the following (left-hand side is Algebraic mode, right-hand side is RPN mode): Creating variables The simplest way to create a variable is by using the K .
Page 2-12 Press ` to create the variable. The variable is now shown in the soft menu key labels: The following are the keystrokes required to enter the remaining variables: A12: 3V5K~a12` Q: ³~„r/„Ü ~„m+~„r™™ K~q` R: „Ô3‚í2‚í1™ K~r` z1: 3+5*„¥ K~„z1` (Accept change to Complex mode if asked).
Page 2-13 • RPN mode (Use H @@OK@@ to change to RPN mode). Use the following keystrokes to store the value of –0.25 into variable α : 0.25` ~‚a` . At this point, the screen will look as follows: This expression means that the value –0.25 is ready to be stored into α .
Page 2-14 p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . The screen, at this point, will look as follows: You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12, α .
Page 2-15 Using the right-shift key followed by soft menu key labels This approach for viewing the contents of a variable works the same in both Algebraic and RPN modes.
Page 2-16 Deleting variables The simplest way of deleting variables is by using function PURGE. This function can be accessed directly by using the TOOLS menu ( I ), or by using the FILES menu „¡ @@OK@@ . Using function PURGE in the stack in Algebraic mode Our variable list contains variables p1, z1, Q, R, and α .
Page 2-17 Using function PURGE in the stack in RPN mode Assuming that our variable list contains the variables p1, z1, Q, R, and α . We will use command PURGE to delete variable p1 .
Page 2-18 this exercise, we use the ORDER command to reorder variables in a directory, we use, in ALG mode: „°˜ Show PROG menu list and select MEMORY @@OK@@ ˜˜˜˜ Show the MEMORY menu list and .
Page 2-19 Press the @CHECK! soft menu key to set flag 117 to soft MENU . The screen will reflect that change: Press @@OK@@ twice to return to normal calculator display. Now, we’ll try to find the ORDER command using similar keystrokes to those used above, i.
Page 2-20 The ORDER command is not shown in this screen. To find it we use the L key to find it: To activate the ORDER command we press the C ( @ORDER ) soft menu key. References For additional information on entering and manipulating expressions in the display or in the Equation Writer see Chapter 2 of the calculator’s user’s guide.
Page 3-1 Chapter 3 Calculations with real numbers This chapter demonstrates the use of the calculator for operations and functions related to real numbers. The user should be acquainted with the keyboard to identify certain functions available in the keyboard (e.
Page 3-2 6.3` 8.5 - 4.2` 2.5 * 2.3` 4.5 / Alternatively, in RPN mode, you can separate the operands with a space ( # ) before pressing the operator key. Examples: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / • Parentheses ( „Ü ) can be used to group operations, as well as to enclose arguments of functions.
Page 3-3 „Ê 2.32` Example in RPN mode: 2.32„Ê • The square function, SQ, is available through „º . Example in ALG mode: „º2.3` Example in RPN mode: 2.3„º The square root function, √ , is available through the R key. When calculating in the stack in ALG mode, enter the function before the argument, e.
Page 3-4 enter the function XROOT followed by the arguments ( y,x ), separated by commas, e.g., ‚»3‚í 27` In RPN mode, enter the argument y , first, then, x , and finally the function call, e.
Page 3-5 2.45` ‚¹ 2.3` „¸ • Three trigonometric functions are readily available in the keyboard: sine ( S ), cosine ( T ), and tangent ( U ). Arguments of these functions are angles in either degrees, radians, grades.
Page 3-6 Real number functions in the MTH menu The MTH ( „´ ) menu include a number of mathematical functions mostly applicable to real numbers. With the default setting of CHOOSE boxes for system flag 117 (see Chapter 2), the M TH menu shows the following functions: The functions are grouped by the type of argument (1.
Page 3-7 For example, in ALG mode, the keystroke sequence to calculate, say, tanh(2.5), is the following: „´4 @@OK@@ 5 @@OK@@ 2.5` In the RPN mode, the keystrokes to perform this calculation are the following: 2.5`„´4 @@OK@@ 5 @@OK@@ The operations shown above assume that you are using the default setting for system flag 117 ( CHOOSE boxes ).
Page 3-8 Finally, in order to select, for example, the hyperbolic tangent (tanh) function, simply press @@TANH@ . Note: To see additional options in these soft menus, press the L key or the „« keystroke sequence.
Page 3-9 Option 1. Tools.. contains functions used to operate on units (discussed later). Options 2. Length.. through 17.Viscosity .. contain menus with a number of units for each of the quantities described. For example, selecting option 8. Force.. shows the following units menu: The user will recognize most of these units (some, e.
Page 3-10 Pressing on the appropriate soft menu key will open the sub-menu of units for that particular selection. For example, for the @) SPEED sub-menu, the following units are available: Pressing the soft menu key @) UNITS will take you back to the UNITS menu.
Page 3-11 Attaching units to numbers To attach a unit object to a number, the number must be followed by an underscore ( ‚Ý , key(8,5)). Thus, a force of 5 N will be entered as 5_N.
Page 3-12 ____________________________________________________ Prefix Name x Prefix Name x ____________________________________________________ Y yotta +24 d deci -1 Z zetta +21 c centi -2 E exa +18 m.
Page 3-13 which shows as 65_(m ⋅ yd). To convert to units of the SI system, use function UBASE (find it using the command catalog, ‚N ): Note: Recall that the ANS(1) variable is available through the keystroke combination „î (associated with the ` key).
Page 3-14 These operations produce the following output: Unit conversions The UNITS menu contains a TOOLS sub-menu, which provides the following functions: CONVERT(x,y): convert unit object x to units.
Page 3-15 The soft menu keys corresponding to this CONSTANTS LIBRARY screen include the following functions: SI when selected, constants values are shown in SI units (*) ENGL when selected, constants .
Page 3-16 To copy the value of Vm to the stack, select the variable name, and press ! ²STK , then, press @QUIT@ . For the calculator set to the ALG, the screen will look like this: The display shows what is called a tagged value , Vm:359.0394 . In here, Vm, is the tag of this result.
Page 3-17 and get the result you want without having to type the expression in the right- hand side for each separate value. In the following example, we assume you have set your calculator to ALG mode.
Page 3-18 between quotes that contain that local variable, and show the evaluated expression. To activate the function in ALG mode, type the name of the function followed by the argument between parentheses, e.
Page 4-1 Chapter 4 Calculations with complex numbers This chapter shows examples of calculations and application of functions to complex numbers. Definitions A complex number z is written as z = x + iy , (Cartesian form) where x and y are real numbers, and i is the imaginary unit defined by i 2 = - 1.
Page 4-2 Entering complex numbers Complex numbers in the calculator can be entered in either of the two Cartesian representations, namely, x+iy , or (x,y) . The results in the calculator will be shown in the ordered-pair format, i.e., (x,y) . For example, with the calculator in ALG mode, the complex number (3.
Page 4-3 The result shown above represents a magnitude, 3.7, and an angle 0.33029…. The angle symbol ( ∠ ) is shown in front of the angle measure. Return to Cartesian or rectangular coordinates by using function RECT (available in the catalog, ‚N ).
Page 4-4 (3+5i) + (6-3i) = (9,2); (5-2i) - (3+4i) = (2,-6) (3-i)(2-4i) = (2,-14); (5-2i)/(3+4i) = (0.28,-1.04) 1/(3+4i) = (0.12, -0.16) ; -(5-3i) = -5 + 3i The CMPLX menus There are two CMPLX (CoMPLeX numbers) menus available in the calculator.
Page 4-5 Examples of applications of these functions are shown next in RECT coordinates. Recall that, for ALG mode, the function must precede the argument, while in RPN mode, you enter the argument first, and then select the function.
Page 4-6 Functions applied to complex numbers Many of the keyboard-based functions and MTH menu functions defined in Chapter 3 for real numbers (e.g., SQ, ,LN, e x , etc.), can be applied to complex numbers. The result is another complex number, as illustrated in the following examples.
Page 4-7 Function DROITE is found in the command catalog ( ‚N ). If the calculator is in APPROX mode, the result will be Y = 5.*(X-5.)-3. Reference Additional information on complex number operations is presented in Chapter 4 of the calculator’s user’s guide.
Page 5-1 Chapter 5 Algebraic and arithmetic operations An algebraic object, or simply, algebraic, is any number, variable name or algebraic expression that can be operated upon, manipulated, and combined according to the rules of algebra. Examples of algebraic objects are the following: • A number: 12.
Page 5-2 After building the object, press ` to show it in the stack (ALG and RPN modes shown below): Simple operations with algebraic objects Algebraic objects can be added, subtracted, multiplied, di.
Page 5-3 In ALG mode, the following keystrokes will show a number of operations with the algebraics contained in variables @@A1@@ and @@A2@@ (press J to recover variable menu): @@A1@@ + @@A2@@ ` @@A1@.
Page 5-4 Functions in the ALG menu The ALG (Algebraic) menu is available by using the keystroke sequence ‚× (associated with the 4 key). With system flag 117 set to CHOOSE boxes , the ALG menu show.
Page 5-5 Copy the examples provided onto your stack by pressing @ECHO! . For example, for the EXPAND entry shown above, press the @ECHO! soft menu key to get the following example copied to the stack (press ` to execute the command): Thus, we leave for the user to explore the applications of the functions in the ALG menu.
Page 5-6 Operations with transcendental functions The calculator offers a number of functions that can be used to replace expressions containing logarithmic and exponential functions ( „Ð ), as well as trigonometric functions ( ‚Ñ ).
Page 5-7 These functions allow to simplify expressions by replacing some category of trigonometric functions for another one. For example, the function ACOS2S allows to replace the function arccosine (acos(x)) with its expression in terms of arcsine (asin(x)).
Page 5-8 FACTORS: SIMP2: The functions associated with the ARITHMETIC submenus: INTEGER, POLYNOMIAL, MODULO, and PERMUTATION, are presented in detail in Chapter 5 in the calculator’s user’s guide. The following sections show some applications to polynomials and fractions.
Page 5-9 The variable VX Most polynomial examples above were written using variable X. This is because a variable called VX exists in the calculator’s {HOME CASDIR} directory that takes, by default, the value of ‘X’. This is the name of the preferred independent variable for al gebraic and calculus applications.
Page 5-10 Note : you could get the latter result by using PARTFRAC: PARTFRAC(‘(X^3-2*X+2)/(X-1)’) = ‘X^2+X-1 + 1/(X-1)’. The PEVAL function The function PEVAL (Polynomial EVALuation) can be us.
Page 5-11 The PROPFRAC function The function PROPFRAC converts a rational fraction into a “proper” fraction, i.e., an integer part added to a fract ional part, if such decomposition is possible.
Page 5-12 The FROOTS function The function FROOTS, in the ARITHMETIC/POLYNOMIAL menu, obtains the roots and poles of a fraction. As an example, applying function FROOTS to the result produced above, will result in: [1 –2. –3 –5. 0 3. 2 1. –5 2.
Page 5-13 Reference Additional information, definitions, and examples of algebraic and arithmetic operations are presented in Chapter 5 of the calculator’s user’s guide.
Page 6-1 Chapter 6 Solution to equations Associated with the 7 key there are two menus of equation-solving functions, the Symbolic SOLVer ( „Î ), and the NUMerical SoLVer ( ‚Ï ). Following, we present some of the functions contained in these menus.
Page 6-2 Using the RPN mode, the solution is accomplished by entering the equation in the stack, followed by the variable, before entering function ISOL.
Page 6-3 The following examples show the use of function SOLVE in ALG and RPN modes (Use Complex mode in the CAS): The screen shot shown above displays two solutions. In the first one, β 4 -5 β =125, SOLVE produces no solutions { }. In the second one, β 4 - 5 β = 6, SOLVE produces four solutions, shown in the last output line.
Page 6-4 Function SOLVEVX The function SOLVEVX solves an equation for the default CAS variable contained in the reserved variable name VX. By default, this variable is set to ‘X’. Examples, using the ALG mode with VX = ‘X’, are shown below: In the first case SOLVEVX could not find a solution.
Page 6-5 To use function ZEROS in RPN mode, enter first the polynomial expression, then the variable to solve for, and then function ZEROS. The following screen shots show the RPN stack before and aft.
Page 6-6 Following, we present applications of items 3. Solve poly.. , 5. Solve finance , and 1. Solve equation.. , in that order. Appendix 1-A, in the calculator’s user’s guide, contains instructions on how to use input forms with examples for the numerical solver applications.
Page 6-7 Press ` to return to stack. The stack will show the following results in ALG mode (the same result would be shown in RPN mode): All the solutions are complex numbers: (0.
Page 6-8 Generating an algebraic expression for the polynomial You can use the calculator to generate an algebraic expression for a polynomial given the coefficients or the roots of the polynomial. The resulting expression will be given in terms of the default CAS variable X.
Page 6-9 Financial calculations The calculations in item 5. Solve finance.. in the Numerical Solver ( NUM.SLV ) are used for calculations of time value of money of interest in the discipline of engineering economics and other financial applications.
Page 6-10 Then, enter the SOLVE environment and select Solve equation… , by using: ‚Ï @@OK@@ . The corresponding screen will be shown as: The equation we stored in variable EQ is already loaded in the Eq field in the SOLVE EQUATION input form. Also, a field labeled x is provided.
Page 6-11 Notice that function MSLV requires three arguments: 1. A vector containing the equations, i.e., ‘[SIN(X)+Y,X+SIN(Y)=1]’ 2. A vector containing the variables to solve for, i.e., ‘[X,Y]’ 3. A vector containing initial values for the solution, i.
Page 6-12 by MSLV is numerical, the information in the upper left corner shows the results of the iterative process used to obtain a solution. The final solution is X = 1.8238, Y = -0.9681 . Reference Additional information on solving single and multiple equations is provided in Chapters 6 and 7 of the calculator’s user’s guide.
Page 7-1 Chapter 7 Operations with lists Lists are a type of calculator’s object that can be useful for data processing. This chapter presents examples of operations with lists. To get started with the examples in this Chapter, we use the Approximate mode (See Chapter 1).
Page 7-2 Addition, subtraction, multiplication, division Multiplication and division of a list by a single number is distributed across the list, for example: Subtraction of a single number from a lis.
Page 7-3 Note : If we had entered the elements in lists L4 and L3 as integers, the infinite symbol would be shown whenever a division by zero occurs. To produce the following result you need to re-ent.
Page 7-4 ABS INVERSE (1/x) Lists of complex numbers You can create a complex number list, say, L5 = L1 ADD i*L2 (type the instruction as indicated before), as follows: Functions such as LN, EXP, SQ, etc., can also be applied to a list of complex numbers, e.
Page 7-5 With system flag 117 set to SOFT menus, the MTH/LIST menu shows the following functions: The operation of the MTH/LIST menu is as follows: ∆ LIST : Calculate increment among consecutive ele.
Page 7-6 The SEQ function The SEQ function, available through the command catalog ( ‚N ), takes as arguments an expression in terms of an index, the name of the index, and starting, ending, and increment values for the index, and returns a list consisting of the evaluation of the expression for all possible values of the index.
Page 8-1 Chapter 8 Vectors This Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as well as physical vectors of 2 and 3 components. Entering vectors In the calculator, vectors are represented by a sequence of numbers enclosed between brackets, and typically entered as row vectors.
Page 8-2 ( ‚í ) or spaces ( # ). Notice that after pressing ` , in either mode, the calculator shows the vector elements separated by spaces. Storing vectors into variables in the stack Vectors can be stored into variables.
Page 8-3 The @EDIT key is used to edit the contents of a selected cell in the Matrix Writer. The @VEC@@ key, when selected, will produce a vector, as opposed to a matrix of one row and many columns. The ← WID key is used to decrease the width of the columns in the spreadsheet.
Page 8-4 The @+ROW@ key will add a row full of zeros at the location of the selected cell of the spreadsheet. The @-ROW key will delete the row corresponding to the selected cell of the spreadsheet. The @+COL@ key will add a column full of zeros at the location of the selected cell of the spreadsheet.
Page 8-5 (3) Move the cursor up two positions by using —— . Then press @-ROW . The second row will disappear. (4) Press @+ROW@ . A row of three zeroes appears in the second row. (5) Press @-COL@ . The first column will disappear. (6) Press @+COL@ .
Page 8-6 Attempting to add or subtract vectors of different length produces an error message: Multiplication by a scalar, and division by a scalar Multiplication by a scalar or division by a scalar is straightforward: Absolute value function The absolute value function (ABS), when applied to a vector, produces the magnitude of the vector.
Page 8-7 The MTH/VECTOR menu The MTH menu ( „´ ) contains a menu of functions that specifically to vector objects: The VECTOR menu contains the following functions (system flag 117 set to CHOOSE boxes): Magnitude The magnitude of a vector, as discussed earlier, can be found with function ABS.
Page 8-8 Cross product Function CROSS (option 3 in the MTH/VECTOR menu) is used to calculate the cross product of two 2-D vectors, of two 3-D vectors, or of one 2-D and one 3- D vector. For the purpose of calculating a cross product, a 2-D vector of the form [A x , A y ], is treated as the 3-D vector [A x , A y ,0].
Page 9-1 Chapter 9 Matrices and linear algebra This chapter shows examples of creating matrices and operations with matrices, including linear algebra applications.
Page 9-2 Press ` once more to place the matrix on the stack. The ALG mode stack is shown next, before and after pressing , once more: If you have selected the textbook display option (using H @) DISP! and checking off Textbook ), the matrix will look like the one shown above.
Page 9-3 Operations with matrices Matrices, like other mathematical objects, can be added and subtracted. They can be multiplied by a scalar, or among themselves. An important operation for linear algebra applications is the inverse of a matrix. Details of these operations are presented next.
Page 9-4 In RPN mode, try the following eight examples: A22 ` B22 `+ A22 ` B22 `- A23 ` B23 `+ A23 ` B23 `- A32 ` B32 `+ A32 ` B32 `- A33 ` B33 `+ A33 ` B33 `- Multiplication There are different multiplication operations that involve matrices. These are described next.
Page 9-5 Matrix multiplication Matrix multiplication is defined by C m × n = A m × p ⋅ B p × n . Notice that matrix multiplication is only possible if the number of columns in the first operand is equal to the number of rows of the second operand.
Page 9-6 The identity matrix The identity matrix has the property that A ⋅ I = I ⋅ A = A . To verify this property we present the following examples using the matrices stored earlier on.
Page 9-7 Characterizing a matrix (The matrix NORM menu) The matrix NORM (NORMALIZE) menu is accessed through the keystroke sequence „´ . This menu is described in detail in Chapter 10 of the calculator’s user’s guide. Some of these functions are described next.
Page 9-8 This system of linear equations can be written as a matrix equation, A n × m ⋅ x m × 1 = b n × 1 , if we define the following matrix and vectors: m n nm n n m m a a a a a a a a a A × .
Page 9-9 . 6 13 13 , , 4 2 2 8 3 1 5 3 2 3 2 1 − − = = − − − = b x A and x x x This system has the same number of equations as of unknowns, and will be referred to as a square system.
Page 9-10 Solution with the inverse matrix The solution to the system A ⋅ x = b , where A is a square matrix is x = A -1 ⋅ b . For the example used earlier, we can find the solution in the calcula.
Page 10-1 Chapter 10 Graphics In this chapter we introduce some of the graphics capabilities of the calculator. We will present graphics of functions in Cartesian coordinates and polar coordinates, parametric plots, graphics of conics, bar plots, scatterplots, and fast 3D plots.
Page 10-2 Plotting an expression of the form y = f(x) As an example, let's plot the function, ) 2 exp( 2 1 ) ( 2 x x f − = π • First, enter the PLOT SETUP environment by pressing, „ô . Make sure that the option Function is selected as the TYPE , and that ‘X’ is selected as the independent variable ( INDEP ).
Page 10-3 VIEW, then press @AUTO to generate the V-VIEW automatically. The PLOT WINDOW screen looks as follows: • Plot the graph: @ERASE @DRAW (wait till the calculator finishes the graphs) • To see labels: @EDIT L @LABEL @MENU • To recover the first graphics menu: LL @) PICT • To trace the curve: @TRACE @@X,Y@@ .
Page 10-4 • We will generate values of the function f(x), defined above, for values of x from –5 to 5, in increments of 0.5. First, we need to ensure that the graph type is set to FUNCTION in the PLOT SETUP screen ( „ô , press them simultaneously, if in RPN mode).
Page 10-5 • • The @ZOOM key, when pressed, produces a menu with the options: In , Out , Decimal, Integer , and Trig . Try the following exercises: • • With the option In highlighted, press @@@OK@@@ . The table is expanded so that the x-increment is now 0.
Page 10-6 • Press „ô , simultaneously if in RPN mode, to access to the PLOT SETUP window. • Change TYPE to Fast3D. ( @CHOOS! , find Fast3D , @@OK@@ ). • Press ˜ and type ‘X^2+Y^2’ @@@OK@@@ . • Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables.
Page 10-7 • When done, press @EXIT . • Press @CANCL to return to the PLOT WINDOW environment. • Change the Step data to read: Step Indep: 20 Depnd: 16 • Press @ERASE @DRAW to see the surface plot. Sample views: • When done, press @EXIT . • Press @CANCL to return to PLOT WINDOW.
Page 10-8 • Press LL @) PICT to leave the EDIT environment. • Press @CANCL to return to the PLOT WINDOW environment. Then, press $ , or L @@@OK@@@ , to return to normal calculator display. Reference Additional information on graphics is available in Chapters 12 and 22 in the calculator’s user’s guide.
Page 11-1 Chapter 11 Calculus Applications In this Chapter we discuss applications of the calculator’s functions to operations related to Calculus, e.
Page 11-2 where the limit is to be calculated. Function lim is available through the command catalog ( ‚N~„l ) or through option 2. LIMITS & SERIES… of the CALC menu (see above). Function lim is entered in ALG mode as lim(f(x),x=a) to calculate the limit ) ( lim x f a x → .
Page 11-3 Anti-derivatives and integrals An anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx. One way to represent an anti-derivative is as a indefinite integral , i.e., C x F dx x f + = ∫ ) ( ) ( if and only if, f(x) = dF/dx, and C = constant.
Page 11-4 Please notice that functions SIGMAVX and SIGMA are designed for integrands that involve some sort of integer f unction like the factorial (!) function shown above. Their result i s the so-called discrete derivative, i.e., one defined for integer numbers only.
Page 11-5 where f (n) (x) represents the n-th derivative of f(x) with respect to x, f (0) (x) = f(x). If the value x 0 = 0, the series is referred to as a Maclaurin’s series. Functions TAYLR, TAYLR0, and SERIES Functions TAYLR, TAYLR0, and SERIES are used to generate Taylor polynomials, as well as Taylor series with residuals.
Page 11-6 expression for h = x - a, if the second argument in the function call is ‘x=a’, i.e., an expression for the increment h. The list returned as the first output object includes the following items: 1 - Bi-directional limit of the function at point of expansion, i.
Page 12-1 Chapter 12 Multi-variate Calculus Applications Multi-variate calculus refers to functions of two or more variables. In this Chapter we discuss basic concepts of mu lti-variate calculus: partial derivatives and multiple integrals.
Page 12-2 To define the functions f(x,y) and g(x,y,z), in ALG mode, use: DEF(f(x,y)=x*COS(y)) ` DEF(g(x,y,z)= √ (x^2+y^2)*SIN(z) ` To type the derivative symbol use ‚ ¿ . The derivative )) , ( ( y x f x ∂ ∂ , for example, will be entered as ∂ x(f(x,y)) ` in ALG mode in the screen.
Page 13-1 Chapter 13 Vector Analysis Applications This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vector analysis.
Page 13-2 Alternatively, use function DERIV as follows: Divergence The divergence of a vector function, F (x,y,z) = f(x,y,z) i +g(x,y,z) j +h(x,y,z) k , is defined by taking a “dot-product” of the del operator with the function, i.e., F divF • ∇ = .
Page 14-1 Chapter 14 Differential Equations In this Chapter we present examples of solving ordinary differential equations (ODE) using calculator functions. A differential equation is an equation involving derivatives of the independent variable. In most cases, we seek the dependent function that satisfies the differential equation.
Page 14-2 Function LDEC The calculator provides function LDEC (Linear Differential Equation Command) to find the general solution to a linear ODE of any order with constant coefficients, whether it is homogeneous or not.
Page 14-3 The solution is: which is equivalent to y = K 1 ⋅ e –3x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + (450 ⋅ x 2 +330 ⋅ x+241)/13500. Function DESOLVE The calculator provides function DESOLVE (Differential Equation SOLVEr) to solve certain types of differential equation s.
Page 14-4 The variable ODETYPE You will notice in the soft-menu key labels a new variable called @ODETY (ODETYPE). This variable is produced with the call to the DESOL function and holds a string showing the type of ODE used as input for DESOLVE. Press @ODETY to obtain the string “ 1st order linear ”.
Page 14-5 Laplace Transforms The Laplace transform of a function f(t) produces a function F(s) in the image domain that can be utilized to find the solution of a linear differential equation involving f(t) through algebraic methods. The steps involved in this application are three: 1.
Page 14-6 and you will notice that the CAS default variable X in the equation writer screen replaces the variable s in this definition. Therefore, when using the function LAP you get back a function of X, which is the Laplace transform of f(X). Example 2 – Determine the inverse Laplace transform of F(s) = sin(s).
Page 14-7 Using the calculator in ALG mode, first we define functions f(t) and g(t): Next, we move to the CASDIR sub-directory under HOME to change the value of variable PERIOD, e.g., „ (hold) §`J @) CASDI `2 K @PERIOD ` Return to the sub-directory where you defined functions f and g, and calculate the coefficients.
Page 14-8 Thus, c 0 = 1/3, c 1 = ( π⋅ i+2)/ π 2 , c 2 = ( π⋅ i+1)/(2 π 2 ). The Fourier series with three elements will be written as g(t) ≈ Re[(1/3) + ( π⋅ i+2)/ π 2 ⋅ exp(i ⋅π⋅ t)+ ( π⋅ i+1)/(2 π 2 ) ⋅ exp(2 ⋅ i ⋅π⋅ t)].
Page 15-1 Chapter 15 Probability Distributions In this Chapter we provide examples of applications of the pre-defined probability distributions in the calculator. The MTH/PROBABILITY.. sub-menu - part 1 The MTH/PROBABILITY.. sub-menu is accessible through the keystroke sequence „´ .
Page 15-2 We can calculate combinations, permutations, and factorials with functions COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of those functions is described next: • COMB.
Page 15-3 The MTH/PROB menu - part 2 In this section we discuss four continuous probability distributions that are commonly used for problems related to statistical inference: the normal distribution, the Student’s t distribution, the Chi-square ( χ 2 ) distribution, and the F-distribution.
Page 15-4 UTPT, given the parameter ν and the value of t, i.e., UTPT( ν ,t) = P(T>t) = 1- P(T<t). For example, UTPT(5,2.5) = 2.7245…E-2. The Chi-square distribution The Chi-square ( χ 2 ) distribution has one parameter ν , known as the degrees of freedom.
Page 16-1 Chapter 16 Statistical Applications The calculator provides the following pre-programmed statistical features accessible through the keystroke combination ‚Ù (the 5 key): Entering data Applications number 1, 2, and 4 from the list above require that the data be available as columns of the matrix Σ DAT.
Page 16-2 The form lists the data in Σ DAT, shows that column 1 is selected (there is only one column in the current Σ DAT). Move about the form with the arrow keys, and press the @ CHK@ soft me.
Page 16-3 Obtaining frequency distributions The application 2. Frequencies.. in the STAT menu can be used to obtain frequency distributions for a set of data. The data must be present in the form of a column vector stored in variable Σ DAT. To get started, press ‚Ù˜ @@@OK@@@ .
Page 16-4 This information indicates that our data ranges from -9 to 9. To produce a frequency distribution we will use the interval (-8,8) dividing it into 8 bins of width 2 each.
Page 16-5 data sets (x,y), stored in columns of the Σ DAT matrix. For this application, you need to have at least two columns in your Σ DAT variable. For example, to fit a linear relationship to the data shown in the table below: x y 0 0.5 1 2.3 2 3.
Page 16-6 Level 3 shows the form of the equation. Level 2 shows the sample correlation coefficient, and level 1 shows the covariance of x-y. For definitions of these parameters see Chapter 18 in the user’s guide. For additional information on the data-fit feature of the calculator see Chapter 18 in the user’s guide.
Page 16-7 • Press @@@OK@@@ to obtain the following results: Confidence intervals The application 6. Conf Interval can be accessed by using ‚Ù— @@@OK@@@ . The application offers the following options: These options are to be interpreted as follows: 1.
Page 16-8 4. Z-INT: p 1− p 2 .: Confidence interval for the difference of two proportions, p 1 -p 2 , for large samples with unknown population variances. 5. T-INT: 1 µ .: Single sample confidence interval for the population mean, µ , for small samples with unknown population variance.
Page 16-9 The graph shows the standard normal distribution pdf (probability density function), the location of the critical points ± z α/2 , the mean value (23.2) and the corresponding interval limits (2 1.88424 and 24.51576). Press @TEXT to return to the previous results screen, and/or press @@@OK@@@ to exit the confidence interval environment.
Page 16-10 1. Z-Test: 1 µ .: Single sample hypothesis testing for the population mean, µ , with known population variance, or for large samples with unknown population variance.
Page 16-11 Select µ ≠ 150 . Then, press @@@OK@@@ . The result is: Then, we reject H 0 : µ = 150 , against H 1 : µ ≠ 150 . The test z value is z 0 = 5.656854. The P-value is 1.54 × 10 -8 . The critical values of ± z α /2 = ± 1.959964, corresponding to critical x range of {147.
Page 17-1 Chapter 17 Numbers in Different Bases Besides our decimal (base 10, digits = 0-9) number system, you can work with a binary system (base 2, digits = 0,1), an octal system (base 8, digits = 0-7), or a hexadecimal system (base 16, digits=0-9,A-F), among others.
Page 17-2 base to be used for binary integers, choose either HEX(adecimal), DEC(imal), OCT(al), or BIN(ary) in the BASE menu. For example, if @HEX ! is selected, binary integers will be a hexadecimal numbers, e.g., #53, #A5B, etc. As different systems are selected, the numbers will be automatically converted to the new current base.
Page 18-1 Chapter 18 Using SD cards The calculator provides a memory card port where you can insert an SD flash card for backing up calculator objects, or for downloading objects from other sources. The SD card in the calculator will appear as port number 3.
Page 18-2 Enter object, type the name of the stored object using port 3 (e.g., :3:VAR1 ), press K . Recalling an object from the SD card To recall an object from the SD card onto the screen, use function RCL, as follows: • In algebraic mode: Press „© , type the name of the stored object using port 3 (e.
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